Department of Pre-University Education, KarnatakaPUC Karnataka Science Class 12
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The Function F ( X ) = Sin ( π [ X − π ] ) 4 + [ X ] 2 , Where [⋅] Denotes the Greatest Integer Function, is (A) Continuous as Well as Differentiable for All X ∈ R - Mathematics

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Question

The function \[f\left( x \right) = \frac{\sin \left( \pi\left[ x - \pi \right] \right)}{4 + \left[ x \right]^2}\] , where [⋅] denotes the greatest integer function, is

Options
  • continuous as well as differentiable for all x ∈ R

  • continuous for all x but not differentiable at some x

  • differentiable for all x but not continuous at some x.

  • none of these

Solution

(a) continuous as well as differentiable for all x ∈ R 

Here, 

\[f\left( x \right) = \frac{\sin \left( \pi\left[ x - \pi \right] \right)}{4 + \left[ x \right]^2}\]

Since, we know that

\[\pi\left[ \left( x - \pi \right) \right] = n\pi\]
\[\ \text { sin n} \pi = 0\]
\[4 + \left[ x \right]^2 \neq 0\]
∴f(x) = 0 for all x

Thus, f(x) is a constant function and it is continuous and differentiable everywhere.

  Is there an error in this question or solution?
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Solution The Function F ( X ) = Sin ( π [ X − π ] ) 4 + [ X ] 2 , Where [⋅] Denotes the Greatest Integer Function, is (A) Continuous as Well as Differentiable for All X ∈ R Concept: Algebra of Continuous Functions.
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